Finite type is affine-local on source and target

Restricting a finite-type ring map to distinguished affine opens localizes the map and retains a finite generating set, so the condition survives affine refinement.

Conversely, let $V=\mathrm{Spec}R$ be an affine target and $U=\mathrm{Spec}A\subseteq f^{-1}(V)$ an affine source open. Around each point of $U$, choose an affine chart $W=\mathrm{Spec}C$ from the given finite-type source cover. First choose a principal open in $U$ contained in $W$, then a smaller principal open in $W$ contained in the first one. The latter is principal in $U$ as well: its defining function on the first open belongs to a localization of $A$, so clearing a power of the first defining function gives one element of $A$. Quasi-compactness of $U$ now supplies a finite cover $U={\bigcup }_{i=1}^n{D}_U(a_i)$ by such common principal opens. Each $A_{a_i}$ is a localization of a finite-type $R$-algebra $C$, hence is finite type over $R$.

Here is the needed finite-localization descent, rather than just an appeal to it. Choose $h_i\in A$ with ${\sum }_i{h}_i{a}_i=1$. For every $i$, choose finitely many $R$-algebra generators of $A_{a_i}$ and write each as $y_{ij}\mathrm{/}{a}_i^{k_{ij}}$ with $y_{ij}\in A$. Let $B\subseteq A$ be the $R$-subalgebra generated by all $a_i,h_i,y_{ij}$. Then $B_{a_i}=A_{a_i}$ for every $i$, and the $a_i$ generate the unit ideal already in $B$. For any $x\in A\mathrm{/}B$, localization gives $a_i^{N_i}x=0$ for some $N_i$ for each $i$. The powers $a_i^{N_i}$ also generate the unit ideal in $B$, so $x=0$. Thus $A=B$ is finitely generated over $R$. This is the finite-type part of Stacks Project, Tag 00EP.

On an affine target $V$, refine any target cover to a finite cover by opens $D_V(r_j)$ that are principal also in the chosen affine target charts, using the common-principal construction of step 2.1. Base change to each $D_V(r_j)$ preserves finite type of the source charts. For an affine $U\subseteq f^{-1}(V)$, the inverse images are the principal opens $D_U(f^{\mathrm{\#}}{r}_j)$; steps 2.1 and 3.1 make their coordinate rings finite type over ${\mathcal{O}}_S(D_V(r_j))$, hence over ${\mathcal{O}}_S(V)$. Applying step 3.1 again gives finite type on $U$. Finally, quasi-compactness of $f$ supplies a finite affine source subcover over each affine target open, so locally finite type plus quasi-compactness is exactly finite type.